Stuart Barber's research page
List of publications
Wavelet methods in Statistics
Wavelets are a class of functions which can be used to generate
orthogonal bases for spaces of functions. Because of the way wavelets
are designed, the description of a "nice" function in terms of a
wavelet basis tends to be very efficient. Here, "efficient" means
that you only need a few wavelets to describe quite complicated
functions.
Wavelets have been used in
engineering, signal processing, and numerical analysis for some time.
In the last fifteen years the statistical community has been applying
wavelets to problems including nonparametric regression, density
estimation, time series analysis, and changepoint detection.
A good website for an introduction to wavelets is Amara's wavelet
page. Another nice document is Wavelets:
Seeing the forest - and the trees, an article written as part of
the National
Academy of Sciences' "Beyond
Discovery" series of articles on important scientific advances.
There are many web sites devoted to wavelets and their use, including
WaveThresh
(help
pages), a free add-on package for wavelet analysis in R or S-Plus .
Also useful are the wavelet
digest and a list of
wavelet researchers.
One of the interesting things about working with wavelets is the fact
that they are used by researchers in many different disciplines. Two
useful introductions to wavelets written from an engineering point of
view are Clemens Valens' "A
Really Friendly Guide to Wavelets" and Robi Polikar's The Engineer's Ultimate Guide to Wavelet Analysis.
WaveBand
There have been many wavelet thresholding rules proposed in the
literature, but relatively little work has been done so far on
producing confidence intervals to go with the curve estimates.
One Bayesian approach is the WaveBand method
described in Barber, Nason &
Silverman (2002). This method has been implemented in WaveThresh;
code and help pages are available on the
WaveBand web page.
Here are some slides for a talk on the
WaveBand method.
CThresh
The best-known wavelets in the statistical literature are the
"extremal phase" and "least asymmetric" families of wavelets due to
Ingrid Daubechies. It is not so well known that there are also
complex-valued Daubechies wavelets (cDws). It turns out that these
complex-valued wavelets are very good at denoising real-valued
signals. Barber & Nason
(2004) describe two denoising techniques that use these cDws.
Code to use these methods in WaveThresh is nearly ready for release.
Sequential Analysis
Sequential methods are of great potential use in clinical trials. The
basic idea is very simple and is based on the fact that during a
clinical trial, data is steadily accumulated. Rather than waiting
until the end of the trial to analyse all the data, periodic
examination of the data accumulated to date means that if there is a
clear difference in favour of one of the treatments the trial can be
stopped earlier than planned. By allowing the possibility of stopping
the trial at an early stage if there is sufficient evidence in favour
of one of the treatments on trial, significant saving in time and
resources can be made. Moreover, it is possible to greatly reduce the
expected number of patients involved in the trial and thus minimise
the number of patients who will be given an inferior treatment.
Software for sequential clinical trials
One of reason that
sequential methods have not seen wider use in clinical trials is the
lack of software designed to implement these methods. Commercial
programs available include S+SeqTrial
(Insightful) and EaSt
(Cytel Software Corporation). However, some authors have also made
programs available over the web. These include
There is also my own code for
designing optimal symmetric and asymmetric group sequential boundaries using the
approach described in Barber
& Jennison (2002).
List of publications
Collaborators
Robert Aykroyd;
Chris Fallaize;
Mark Gilthorpe;
Alex Goodwin;
Richard Jackson;
Christopher Jennison;
John Kent;
Kanti Mardia;
Guy Nason;
Sam Peck;
Bernard Silverman;
Rebecca Walls.
[University of Leeds]
[School of Mathematics]
[Department of Statistics]
[Home page]
Last modified: Wed Jul 30 16:44:36 BST 2008